Preferred traces on C*-algebras of self-similar groupoids arising as fixed points

TL;DR

This paper studies fixed points of trace-preserving self-mappings on C*-algebras arising from self-similar groupoid actions, showing existence and uniqueness of a universal attractor trace that extends to a KMS state.

Contribution

It proves the existence and uniqueness of a fixed point trace for self-similar groupoid actions, linking it to KMS states on the associated C*-algebra.

Findings

Each self-mapping has a unique fixed point trace.

The fixed point is a universal attractor.

The fixed point trace extends to a KMS state.

Abstract

Recent results of Laca, Raeburn, Ramagge and Whittaker show that any self-similar action of a groupoid on a graph determines a 1-parameter family of self-mappings of the trace space of the groupoid C*-algebra. We investigate the fixed points for these self-mappings, under the same hypotheses that Laca et al. used to prove that the C*-algebra of the self-similar action admits a unique KMS state. We prove that for any value of the parameter, the associated self-mapping admits a unique fixed point, which is in fact a universal attractor. This fixed point is precisely the trace that extends to a KMS state on the C*-algebra of the self-similar action.